The effective geometry Monte Carlo algorithm: Applications to molecular communication

In this work, we address the systematic biases and random errors stemming from finite step sizes encountered in diffusion simulations. We introduce the Effective Geometry Monte Carlo (EG-MC) simulation algorithm which modifies the geometry of the receiver. We motivate our approach in a 1D toy model and then apply our findings to a spherical absorbing receiver in a 3D unbounded environment. We show that with minimal computational cost the impulse response of this receiver can be precisely simulated using EG-MC. Afterwards, we demonstrate the accuracy of our simulations and give tight constraints on the single free parameter in EG-MC. Finally, we comment on the range of applicability of our results. While we present the EG-MC algorithm for the specific case of molecular diffusion, we believe that analogous methods with effective geometry manipulations can be utilized to approach a variety of problems in other branches of physics such as condensed matter physics and cosmological large scale structure simulations.     

Periods of generalized Fermat curves

Let $k,n\ge 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface S that admits a subgroup of conformal automorphisms $H\le \text{Aut}(S)$ isomorphic to ${\mathbb{Z}}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\stackrel{?}{\mathbb{C}}$ and has $n+1$ branch points, each one of order k. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.     

Symmetry and the golden ratio in the analysis of a regular pentagon

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.     

Hyperinstantons,the Beltrami equation,and triholomorphic maps

We consider the Beltrami equation for hydrodynamics and we show that its solutions can be viewed as instanton solutions of a more general system of equations. The latter are the equations of motion for an sigma model on 4‐dimensional worldvolume (which is taken locally HyperKähler) with a 4‐dimensional HyperKähler target space. By means of the 4D twisting procedure originally introduced by Witten for gauge theories and later generalized to 4D sigma‐models by Anselmi and Fré, we show that the equations of motion describe triholomophic maps between the worldvolume and the target space. Therefore, the classification of the solutions to the 3‐dimensional Beltrami equation can be performed by counting the triholomorphic maps. The counting is easily obtained by using several discrete symmetries. Finally, the similarity with holomorphic maps for sigma on Calabi‐Yau space prompts us to reformulate the problem of the enumeration of triholomorphic maps in terms of a topological sigma model.     

Homoclinic bifurcation for a general state-dependent Kolmogorov type predator–prey model with harvesting

In this paper, a general Kolmogorov type predator–prey model is considered. Together with a constant-yield predator harvesting, the state dependent feedback control strategies which take into account the impulsive harvesting on predators as well as the impulsive stocking on the prey are incorporated in the process of population interactions. We firstly study the existence of an order-1 homoclinic cycle for the system. It is shown that an order-1 positive periodic solution bifurcates from the order-1 homoclinic cycle through a homoclinic bifurcation as the impulsive predator harvesting rate crosses some critical value. The uniqueness and stability of the order-1 positive periodic solution are derived by applying the geometry theory of differential equations and the method of successor function. Finally, some numerical examples are provided to illustrate the main results. These results indicate that careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts.     

Infinitely many singular interactions on noncompact manifolds

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).     

Representation invariant Geometrothermodynamics: Applications to ordinary thermodynamic systems

In this work we employ a recently devised metric within the Geometrothermodynamics program to study ordinary thermodynamic systems. The new feature of this metric is that, in addition to Legendre symmetry, it exhibits invariance under a change of representation. This metric was derived in a previous work by the authors while addressing the problem of the conformal structure of the thermodynamic metrics for different representations. Here, we present a thorough analysis for the ideal gas, the van der Waals fluid, the one dimensional Ising model and some other systems of cosmological interest.     

Noether's Theorem on Surfaces

In this article we prove a version of Noether's Theorem (of Calculus of Variations) which is valid for a general regular (compact) surface. As a special feature, the Lie group of transformations is allowed to act on the Cartesian product of the surface and the functional space. Additionally, we apply the Theorem to a problem in Classical Differential Geometry of surfaces. The given application is actually an example showing how Noether's Theorem can be used to construct invariant properties of the solutions to variational problems defined on surfaces, or equivalently, of the solutions to the associated Euler-Lagrange equations resulting from them.     

The role of multiple solution tasks in developing knowledge and creativity in geometry

This paper describes changes in students’ geometrical knowledge and their creativity associated with implementation of Multiple Solution Tasks (MSTs) in school geometry courses. Three hundred and three students from 14 geometry classes participated in the study, of whom 229 students from 11 classes learned in an experimental environment that employed MSTs while the rest learned without any special intervention in the course of one school year. This longitudinal study compares the development of knowledge and creativity between the experimental and control groups as reflected in students’ written tests. Geometry knowledge was measured by the correctness and connectedness of the solutions presented. The criteria for creativity were: fluency, flexibility, and originality. The findings show that students’ connectedness as well as their fluency and flexibility benefited from implementation of MSTs. The study supports the idea that originality is a more internal characteristic than fluency and flexibility, and therefore more related with creativity and less dynamic. Nevertheless, the MSTs approach provides greater opportunity for potentially creative students to present their creative products than conventional learning environment. Cluster analysis of the experimental group identified three clusters that correspond to three levels of student performance, according to the five measured criteria in pre- and post-tests, and showed that, with the exception of originality, performance in all three clusters generally improved on the various criteria.